ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.13(2012) No.1,pp.94-98 Exact Solutions of a Generalized KdV-mKdV Equation Cesar A. G ´ omez Sierra 1 , Motlatsi Molati 2 ∗ , Motlatsi P. Ramollo 2 1 Department of Mathematics, Universidad Nacional de Colombia, Bogot´ a Colombia. 2 Department of Mathematics and Computer Science, National University of Lesotho, PO Roma 180, Lesotho. (Received 26 January 2011 , accepted 9 August 2011) Abstract : We study the combined Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equa- tions. By using symmetries we construct the exact solutions for the underlying equation. Exact solutions in explicit form are obtained for two particular cases. Other soliton solutions are derived by using the improved tanh-coth method. Keywords : KdV-mKdV equation ; Lie point symmetry ; exact solution ; solitons 1 Introduction In practical applications finding the exact solutions for nonlinear partial differential equations (NLPDEs) is one of the most important task. Until this end, several direct and computational methods have been implemented. Among the direct methods often used are the Hirota method [1], inverse scattering method [2], and analysis by Lie group theory [3],[4], [5]. On the other hand, the tanh-coth [6] method is one of the frequently used computational method. The combined KdV equation and modified KdV equation (KdV-mKdV) [7] that we study reads u t + auu x +¯ au 2 u x + bu xxx =0, (1) with constants a, ¯ a and b. Particular cases of (1) and similar models have been studied extensively using various approaches [8],[9] and the references therein. The main goal in this work is obtain the traveling wave solutions to (1) using two approaches. First, we use Lie point symmetries of the underlying equation to obtain the most general traveling wave solution and then we consider explicit solutions for two particular cases of (1). Thereafter, we use the improved tanh-coth method to obtain other exact solutions to (1). The paper is organized as follows : In Sec. 2, we use the Lie symmetry approach to derive the traveling wave solution to (1) in an implicit form. As a consequence the soliton solution for two particular cases are analyzed. In Sec. 3, we use the improved tanh-coth method to obtain the exact traveling wave solutions to (1) in explicit form. Finally, some conclusions are given. 2 Lie symmetry approach According to Lie’s algorithm [3],[4], [5], the infinitesimal generator of the maximal symmetry group admitted by (1) is given by X = ξ 1 (t, x, u) ∂ ∂t + ξ 2 (t, x, u) ∂ ∂x + η(t, x, u) ∂ ∂u , (2) if and only if the invariance condition of (1) is X [3] ( u t + auu x +¯ au 2 u x + bu xxx ) =0, (3) where X [3] = X + ζ t ∂ ∂u t + ζ x ∂ ∂u x + ζ xxx ∂ ∂u xxx , (4) ∗. Corresponding author. E-mail address : m.molati@gmail.com Copyright c ⃝World Academic Press, World Academic Union IJNS.2012.02.15/580